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ブートストラップ方程式の自動生成と臨界現象の数値的研究への応用

Mang Chiu Ng

Published Mar 23, 2020• 0 citations
Publisher

Abstract

In this thesis, we introduce new tools for the conformal bootstrap, autoboot and qboot.Each tool solves a different step in the whole computational stack, and combined with an existing efficient tool SDPB which solves semidefinite programming (SDP), our tools make it easier to study conformal field theories with more complicated global symmetries and with more general spectra.In the introduction, we review how the conformal bootstrap method gives rich information about the theory at the fixed point of renormalization group, or in other words, the critical phenomena such as the Ising model at criticality.The following three sections focus on the theories behind the implementation of autoboot and qboot, and the explicit implementation, freely available at https://github.com/selpoG/autoboot/and https://github.com/selpoG/qboot/, is discussed in section 5. We also have two applications in the last section, the Ising model and the O(2) vector model in three dimensions, each of them has close relationship with a physical system in the real world.

DOI
10.15083/0002004682

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In this thesis, we introduce new tools for the conformal bootstrap, autoboot and qboot.Each tool solves a different step in the whole computational stack, and combined with an existing efficient tool SDPB which solves semidefinite programming (SDP), our tools make it easier to study conformal field theories with more complicated global symmetries and with more general spectra.In the introduction, we review how the conformal bootstrap method gives rich information about the theory at the fixed point of renormalization group, or in other words, the critical phenomena such as the Ising model at criticality.The following three sections focus on the theories behind the implementation of autoboot and qboot, and the explicit implementation, freely available at https://github.com/selpoG/autoboot/and https://github.com/selpoG/qboot/, is discussed in section 5. We also have two applications in the last section, the Ising model and the O(2) vector model in three dimensions, each of them has close relationship with a physical system in the real world.
These lectures were given at the Weizmann Institute in the spring of 2019. They are intended to familiarize students with the nuts and bolts of the numerical bootstrap as efficiently as possible. After a brief review of the basics of conformal field theory in d>2 spacetime dimensions, we discuss how to compute conformal blocks, formulate the crossing equations as a semi-definite programming problem, solve this problem using SDPB on a personal computer, and interpret the results. We include worked examples for all steps, including bounds for 3d CFTs with Z2 or O(N) global symmetries. Each lecture includes a problem set, which culminate in a precise computation of the 3d Ising model critical exponents using the mixed correlator Z2 bootstrap. A Mathematica file is included that transforms crossing equations into the proper input form for SDPB.
We introduce the use of reinforcement-learning (RL) techniques to the conformal-bootstrap program.We demonstrate that suitable soft Actor-Critic RL algorithms can perform efficient, relatively cheap highdimensional searches in the space of scaling dimensions and OPE-squared coefficients that produce sensible results for tens of CFT data from a single crossing equation.In this paper we test this approach in well-known 2D CFTs, with particular focus on the Ising and tricritical Ising models and the free compactified boson CFT.We present results of as high as 36-dimensional searches, whose sole input is the expected number of operators per spin in a truncation of the conformal-block decomposition of the crossing equations.Our study of 2D CFTs uses only the global soð2; 2Þ part of the conformal algebra, and our methods are equally applicable to higher-dimensional CFTs.When combined with other, already available, numerical and analytical methods, we expect our approach to yield an exciting new window into the nonperturbative structure of arbitrary (unitary or nonunitary) CFTs.
This thesis explores the application of numerical bootstrap techniques to two-point functions of bulk insertions in the presence of a boundary in two-dimensional conformal field theories. The first part of this thesis is devoted to presenting a comprehensive introduction to essential concepts in the numerical bootstrap of two-dimensional CFTs. Namely, an extensive introduction to the basics of conformal field theories (CFTs), their properties arising from the algebraic structure of conformal transformations, consitency conditions on the observables in a CFT, as well as the effects of the presence of defects in two-dimensional CFTs, is given. This is followed by a quick recount of the numerical techniques employed in the literature to solve for the aforementioned consistency conditions. These chapters form an overview of the seminal works that set the backdrop of the contributions made in the second part of the thesis. In the contributions of the thesis, we discuss the progress made in the investigations in (Meineri and Radhakrishnan 2025) where two-point functions in BCFTs are bootstrapped by treating them as a part of a larger system of consistency conditions involving the partition function on an annulus, the two-point function in the presence of a boundary, and four-point functions on the infinite plane. This perspective of treating the two-point function as a part of a larger system makes accessible parts of the BCFT data that have so-far remained inaccessible, and we present results on the boundary gap in the two-point function, and improvements in existing numerical bounds on the boundary entropy.
Two-dimensional Conformal Field Theories (CFTs) allow us to describe quantum field theories at critical points and in general systems without a characteristic scale. It is possible to implement such theories on topologically non-trivial surfaces. If we formulate a two-dimensional CFT on a toroidal surface, a new symmetry emerges known as modular invariance. This symmetry corresponds to the set of transformations that act without changing the toroidal surface, consequently, the partition function of the theory must remain invariant under these transformations. This invariance is used as a constraint by the modular bootstrap method to obtain information about the spectrum without using the Lagrangian and operator content of the theory. By imposing that the theory is consistent with modular symmetry, it is possible to determine the limits of the spectrum beyond which modular invariance can no longer be satisfied, thus constraining the space of admissible two-dimensional conformal field theories. To study CFTs that present discrete global symmetries, it is essential to introduce Topological Defect Lines (TDLs). These are one-dimensional, topologically invariant operators that implement the action of the symmetry group on the Hilbert space. The presence of TDLs modifies the topology of the system and introduces new partition functions associated with different symmetry sectors. Under modular transformations, the TDLs mix among themselves, and consequently, the corresponding partition functions also mix, thus promoting the modular constraint to a vector-valued condition. This formalism makes it possible to implement discrete symmetries also considering the possible presence of anomalies. The objective of this thesis is therefore to understand how discrete symmetries and anomalies affect the spectrum of a CFT. Based on the modular bootstrap framework developed by Lin and Shao for the Z2 symmetry, the analysis was first analytically extended to the Abelian case Z3. This was useful in extending the discussion to a more complex case, namely the non-Abelian symmetry S3. This extension constitutes the original theoretical contribution of the thesis, providing the explicit formulation of modular bootstrap equations for CFTs with non-Abelian discrete symmetry. The analytical modular constraints were reformulated numerically, bridging the modular bootstrap framework with semidefinite programming in SDPB. This analysis paves the way for a numerical study based on convex optimization techniques to explore the constraints arising from modular bootstrap equations.
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Conformal field theories (CFTs) and supersymmetric field theories constitute special classes of quantum field theories with applications ranging from critical phenomena and confinement to quantum gravity via holography. Their large symmetry groups make them ideal testing grounds for theoretical developments, enabling the use of powerful and elegant methods. Among these, conformal bootstrap and integrability have proven particularly successful. In this thesis, we take steps toward extending these methods beyond their conventional domains by breaking conformal symmetry through finite temperature effects and reducing supersymmetry via orbifolding. For spin chains arising from N=2\mathcal{N} = 2N=2 superconformal field theories obtained by orbifolding N=4\mathcal{N} = 4N=4 super Yang–Mills theory in four dimensions, we generalize the coordinate Bethe ansatz by incorporating long-range corrections.On the conformal bootstrap side, after breaking conformality by placing the CFT on a d-dimensional cylinder, we develop an analytic framework based on dispersion relations. We apply this method to several theories, including large-N CFTs in four dimensions. Furthermore, we combine this analytic approach with existing numerical bootstrap techniques at finite temperature to compute new thermal correlation functions in the three-dimensional Ising CFT.
Statistical systems near a classical critical point have been intensively studied from both theoretical and experimental points of view. In particular, correlation functions are of relevance in comparing theoretical models with the experimental data of real systems. In order to compute physical quantities near a critical point, one needs to know the model at the critical (conformal) point. In this line, recent progress in the knowledge of conformal field theories, through the conformal bootstrap, gives the hope of getting some interesting results also outside of the critical point. In this paper, we will review and clarify how, starting from the knowledge of the critical correlators, one can calculate in a safe way their behavior outside the critical point. The approach illustrated requires the model to be just scale invariant at the critical point. We will clarify the method by applying it to different kind of perturbations of the 2D Ising model.